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Question: R :{ 1, 2, 3}\( \to \) {1,2,3} Given by \(R = \\{ (1,1),(2,2),(3,3),(1,2)\\} \). Check if \(R\) is ...

R :{ 1, 2, 3}→\to {1,2,3} Given by R=(1,1),(2,2),(3,3),(1,2)R = \\{ (1,1),(2,2),(3,3),(1,2)\\} . Check if RR is
(a) Reflexive
(b) Symmetric

Explanation

Solution

To check for reflexive and symmetric we will move with the definition of reflexive and symmetric relation definition. Reflexive relation defined as if aa belong to the domain of relation R then (a, a) should be in relation R for every aa belongs to the domain of R. For symmetric relation it states that if (a, b) belong to relation R such that aa belongs to the domain of R and bb belongs to co-domain of R then (b, a) should belong to R such that b belongs to the domain of R and aa belong to co-domain of R.

Complete step by step solution:
Domain of Relation R =1,2,3\\{ 1,2,3\\} and Co-domain of relation R =1,2,3= \\{ 1,2,3\\} and given: R=(1,1),(2,2),(3,3),(1,2)R = \\{ (1,1),(2,2),(3,3),(1,2)\\}

For Reflexive: (a,a)∈R,∀a∈(a,a) \in R,\forall a \in Domain of RR Here, in R,      (1,1)(2,2)(3,3)R,\;\;\;{\kern 1pt} \\{ (1,1)(2,2)(3,3)\\} all are present means for 1∈1,2,31 \in 1,2,3 there is (1,1)∈R(1,1) \in R. Similarly (2,2) and (3,3) belongs to R. Hence, it satisfies the reflexive relation.
Therefore, R is a reflexive relation.

For Symmetric: if (a,b)∈R(a,b) \in R such that a∈a \in Domain of R and b∈b \in Co -domain of R then (b,a)∈R\left( {b,a} \right) \in R where b∈b \in Domain of R, and a ∈\in co-domain in R⋅R \cdot
(1,1)→(1,1) \to (1,1) [True]
(2,2)→(2,2) \to (2,2) [True]
(3,3)→(3,3) \to (3,3) [True]
but (1,2) does not have (2,1) in R .Hence, it is not Symmetric

Given RR is Reflexive but not symmetric.

Note:
A relation between two sets is a collection of ordered pairs containing one object from each set. If the object x is from the first set and the object y is from the second set, then the objects are said to be related if the ordered pair (x, y) is in the relation. A function is a type of relation.